A124973 -id:A124973 - cp's OEIS frontend

A000992 "Half-Catalan numbers": a(n) = Sum_{k=1..floor(n/2)} a(k)*a(n-k) with a(1) = 1.

1, 1, 1, 2, 3, 6, 11, 24, 47, 103, 214, 481, 1030, 2337, 5131, 11813, 26329, 60958, 137821, 321690, 734428, 1721998, 3966556, 9352353, 21683445, 51296030, 119663812, 284198136, 666132304, 1586230523, 3734594241, 8919845275, 21075282588, 50441436842

Offset: 1

N. J. A. Sloane

From David Callan, Nov 02 2006: (Start)
a(n) = number of (unlabeled, rooted) ordered trees on n-1 vertices in which all outdegrees are <= 2 and, for each vertex of outdegree 2, the sizes of its two subtrees are weakly increasing left to right (n >= 2). The number b(n) of such trees on n vertices satisfies the recurrence b[1]=1; b[n_]/;n>=2 := b[n] = b[n-1] + Sum_{i=1..floor((n-1)/2)} b[i]b[n-1-i], the first term counting trees whose root has outdegree 1 and the sum counting trees whose root has outdegree 2 by size of the left subtree. This recurrence generates b(n) = a(n+1), n >= 1. For example, the a(5)=3 such trees are:
.|....|...../\..
.|.../.\.....|..
.|.............. (End)
From R. J. Mathar, Mar 27 2009: (Start)
The connection with the Rayleigh polynomials Phi(2n,x) of A158616 is that Phi(2n,x) = Sum_{i=1..a(n)} 2^(n_i) Product_{j=2..n-1} (x+j)^(n_ij), as described by Kishore.
So a(n) counts the terms in the representation of the polynomial Phi(2n,x) as a sum over these "base" polynomials.
For example, Phi(12,x) = 2^4*(x+2)^2*(x+3) + 2^2*(x+2)*(x+3)^2 + 2^3*(x+2)*(x+3)*(x+4) + 2^3*(x+2)*(x+3)*(x+5) + 2^2*(x+2)*(x+4)*(x+5) + 2*(x+3)^2*(x+5) has a(6)=6 terms. (End)
From Wolfdieter Lang, Jan 06 2012: (Start)
The o.g.f. G(x) := Sum_{n>=0} a(n)*x^n, with a(0)=0, satisfies the relation (G(x))^2 - 2*G(x) + G2(x^2) + 2*x = 0, with the o.g.f. G2(x) := Sum_{n>=0} a(n)^2*x^n of the squares. This can be proved from the connection to the half-convolution of the sequence with itself (for this notion see a comment on A201204, where also the rule for the o.g.f. is given). (End)
Limit_{n->infinity} a(n)^(1/n) = 2.49086422... . - Vaclav Kotesovec, Oct 15 2014
This sequence diverges from A001190 for n >= 8. A001190(n) gives the number of unlabeled binary trees with n leaves and n-1 internal nodes. - Andrew Howroyd, Apr 01 2023

			G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 24*x^8 + 47*x^9 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..2500 (first 200 terms from T. D. Noe)
E. Bohl, P. Lancaster, Implementation of a Markov model for phylogenetic trees, J. Theor. Biol. 239 (3) (2006) 324-333.
J. T. Butler, Letter to N. J. A. Sloane, Jun. 1975.
David S. Evans, Stars of Higher Multiplicity, Quarterly Journal of the Royal Astronomical Society, Vol. 9 (1968), pp. 388-400.
T. Feil, K.Hutson, R. J. Kretchmar, Tree traversals and permutations, Congr. Numer. 175 (2005) 201-221 (mentions the sequence only in the references, not in the text).
N. Kishore, A structure of the Rayleigh polynomial, Duke Math. J., 31 (1964), 513-518.

Compare recurrence for A000108 (the Catalan numbers).
A093637 counts above trees without the restriction that all outdegrees are <= 2.
Cf. A001190, A124973, A248748.

a000992 n = a000992_list !! (n-1)
a000992_list = 1 : f 1 0 [1] where
   f x y zs = z : f (x + y) (1 - y) (z:zs) where
     z = sum $ take x $ zipWith (*) zs $ reverse zs
-- Reinhard Zumkeller, Dec 21 2011

al := 1/2; M1 := 30; a[ 0 ] := 1; for n from 0 to M1 do n0 := floor(al*n);
a[ n+1 ] := sum( a[ i ]*a[ n-i ], i=0..n0); i := 'i'; od: [ seq(a[ j ],j=0..M1) ];
# second Maple program:
a:= proc(n) option remember; `if`(n=1, 1,
      add(a(j)*a(n-j), j=1..n/2))
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Sep 22 2019

a[1]=1; a[n_]:=a[n]=Sum[a[k] a[n-k],{k,1,Floor[n/2]}]; Table[a[n],{n,1,32}] (* Jean-François Alcover, Mar 21 2011 *)

A000992_list(n)={for(i=4,#n=vector(n,i,1),n[i]=sum(j=1,i\2,n[j]*n[i-j]));n}  \\ M. F. Hasler, Dec 20 2011

from functools import lru_cache
@lru_cache(maxsize=None)
def A000992(n): return sum(A000992(k)*A000992(n-k) for k in range(1,(n>>1)+1)) if n>1 else 1 # Chai Wah Wu, Nov 04 2024

A348850 a(n) is the number of labeled rooted unordered binary trees T where the nodes are labeled with distinct positive integers, the root has label n, each parent label equals the sum of its children labels, and T cannot be extended.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 5, 9, 10, 14, 22, 33, 57, 66, 94, 132, 188, 317, 454, 576, 806, 1083, 1535, 2342, 3215, 5231, 5656, 8545, 10804, 15226, 21153, 30342, 44536, 63165, 73877, 107241, 133994, 178497, 247564, 331695, 472331

Offset: 1

Rémy Sigrist, Nov 01 2021

For any n > 0:
- we can imagine a variant of Grundy's game where we start with n at root position,
- and each move consists in adding to a leaf, say w, two children, u and v such that 0 < u < v and u+v = w and u and v do not already appear in the tree,
- a(n) gives the number of final positions (where no move is possible).

			For n = 1, 2, 3, 4: a(n) = 1:
          |         |         |         |
          1         2         3         4
                             / \       / \
                            1   2     1   3
For n = 5, 6: a(n) = 2:
          |         |         |         |
          5         5         6         6
         / \       / \       / \       / \
        1   4     2   3     1   5     2   4
                               / \       / \
                              2   3     1   3

Rémy Sigrist, PARI program for A348850
Wikipedia, Grundy's game

Cf. A124973, A346523, A348851.

PARI
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See Links section.
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A178833 Partial sums of "Half-Catalan numbers" A000992.

Original entry on oeis.org

1, 2, 3, 5, 8, 14, 25, 49, 96, 199, 413, 894, 1924, 4261, 9392, 21205, 47534, 108492, 246313, 568003, 1302431, 3024429, 6990985, 16343338, 38026783, 89322813, 208986625, 493184761, 1159317065, 2745547588, 6480141829, 15399987104, 36475269692, 86916706534, 206503331542

Offset: 1

Jonathan Vos Post, Jan 01 2011

The subsequence of primes begins: 2, 3, 5, 199, 4261, 493184761.

The subsequence of perfect powers begins: 1, 8, 25, 49.

			A000992 starts with 1, 1, 1, 2, 3, ... giving partial sums 1, 2, 3, 5, 8 ...

Cf. A000992, A001190, A093637, A124973.

b:= proc(n) option remember; `if`(n=1, 1,
      add(b(j)*b(n-j), j=1..n/2))
    end:
a:= proc(n) option remember; `if`(n<1, 0, b(n)+a(n-1)) end:
seq(a(n), n=1..42);  # Alois P. Heinz, Nov 04 2024

lista(nn) = for (k=1, nn, print1(vecsum(A000992_list(k)), ", ")); \\ Michel Marcus, Feb 16 2015

a(n) = Sum_{i=1..n} A000992(i).

cp's OEIS Frontend

A000992 "Half-Catalan numbers": a(n) = Sum_{k=1..floor(n/2)} a(k)*a(n-k) with a(1) = 1.

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A348850 a(n) is the number of labeled rooted unordered binary trees T where the nodes are labeled with distinct positive integers, the root has label n, each parent label equals the sum of its children labels, and T cannot be extended.

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A178833 Partial sums of "Half-Catalan numbers" A000992.

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Programs

Maple

PARI

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